Continuous Convex Research Articles

First and Second-Order Approximations as Derivatives of Mappings in Optimality Conditions for Nonsmooth Vector Optimization

First and second-order approximations are used to establish both necessary and sufficient optimality conditions for local weak efficiency and local firm efficiency in nonsmooth set-constrained vector problems. Even continuity and relaxed convexity assumptions are not imposed. Compactness conditions are also relaxed. Examples are provided to show advantages of the presented results over recent existing ones.

Continuous convexity and canonical partial metrics in normed spaces

Continuous convexity and canonical partial metrics in normed spaces

Restricted Weak Upper Semi-continuity of Subdifferentials of Convex Functions on Banach Spaces

Let X be a Banach space and f a continuous convex function on X. Suppose that for each x ∈ X and each weak neighborhood V of zero in X* there exists δ > 0 such that $$\partial f(y)\subset\partial f(x)+V\;\;{\rm for\;all}\;y\in X\;{\rm with}\;\|y-x\|<\delta. $$ Then every continuous convex function g with \(g \leqslant f\) on X is generically Frechet differentiable. If, in addition, \(\lim\limits_{\|x\|\rightarrow\infty}f(x)=\infty\), then X is an Asplund space.

Delta-semidefinite and Delta-convex Quadratic Forms in Banach Spaces

A continuous quadratic form (“quadratic form”, in short) on a Banach space X is: (a) delta-semidefinite (i.e., representable as a difference of two nonnegative quadratic forms) if and only if the corresponding symmetric linear operator $$T: X\rightarrow X^*$$ factors through a Hilbert space; (b) delta-convex (i.e., representable as a difference of two continuous convex functions) if and only if T is a UMD-operator. It follows, for instance, that each quadratic form on an infinite-dimensional L p (μ) space (1 ≤ p ≤ ∞) is: (a) delta-semidefinite iff p ≥ 2; (b) delta-convex iff p > 1. Some other related results concerning delta-convexity are proved and some open probms are stated.

Characterizations of convex functions of a vector variable via Hermite-Hadamard's inequality

The classical Hermite-Hadamard inequality characterizes the continuous convex func- tions of one real variable. The aim of the present paper is to give an analogous characterization for functions of a vector variable. 1. The Hermite-Hadamard inequality

Quotients of Continuous Convex Functions on Nonreflexive Banach Spaces

On each nonreflexive Banach space $X$ there exists a positive continuous convex function $f$ such that $1/f$ is not a d.c. function (i.e., a difference of two continuous convex functions). This result together with known ones implies that $X$ is refl

On sets of non-differentiability of Lipschitz and convex functions

We observe that each set from the system $\widetilde{\mathcal A}$ (or even $\widetilde{\mathcal{C}}$) is $\Gamma $-null; consequently, the version of Rademacher’s theorem (on Gâteaux differentiability of Lipschitz functions on separable Banach spaces) proved by D. Preiss and the author is stronger than that proved by D. Preiss and J. Lindenstrauss. Further, we show that the set of non-differentiability points of a convex function on ${\mathbb{R}}^n$ is $\sigma $-strongly lower porous. A discussion concerning sets of Fréchet non-differentiability points of continuous convex functions on a separable Hilbert space is also presented.

A characterization of B-convexity and J-convexity of Banach spaces

In [K.-I. Mitani and K.-S. Saito, J. Math. Anal. Appl. 327 (2007), 898-907] we characterized the strict convexity, uniform convexity and uniform non-squareness of Banach spaces using $\psi$-direct sums of two Banach spaces, where $\psi$ is a continuous convex function with some appropriate conditions on $[0,1]$. In this paper, we characterize the $B_n$-convexity and $J_n$-convexity of Banach spaces using $\psi$-direct sums of $n$ Banach spaces, where $\psi$ is a continuous convex function with some appropriate conditions on a certain convex subset of $\mathbb R^n$.

Asymptotics for some nonlinear damped wave equation: finite time convergence versus exponential decay results

Given a bounded open set Ω⊂Rn and a continuous convex function Φ:L2(Ω)→R, let us consider the following damped wave equation(S)utt−Δu+∂Φ(ut)∋0,(t,x)∈(0,+∞)×Ω, under Dirichlet boundary conditions. The notation ∂Φ refers to the subdifferential of Φ in the sense of convex analysis. The nonlinear term ∂Φ allows to modelize a large variety of friction problems. Among them, the case Φ=|⋅|L1 corresponds to a Coulomb friction, equal to the opposite of the velocity sign. After we have proved the existence and uniqueness of a solution to (S), our main purpose is to study the asymptotic properties of the dynamical system (S). In two significant situations, we bring to light an interesting phenomenon of dichotomy: either the solution converges in a finite time or the speed of convergence is exponential as t→+∞. We also give conditions which ensure the finite time stabilization of (S) toward some stationary solution.

A simple proof of inequalities of integrals of composite functions

In this paper we give a simple proof of inequalities of integrals of functions which are the composition of nonnegative continuous convex functions on a vector space R m and vector-valued functions in a weakly compact subset of a Banach vector space generated by m L μ p -spaces for 1 ⩽ p < + ∞ . Also, the same inequalities hold if these vector-valued functions are in a weakly* compact subset of a Banach vector space generated by m L μ ∞ -spaces instead.

Characterizations of the Solution Sets of Convex Programs and Variational Inequality Problems

For a convex program in a normed vector space with the objective function admitting the Gâteaux derivative at an optimal solution, we show that the solution set consists of the feasible points lying in the hyperplane whose normal vector equals the Gâteaux derivative. For a general continuous convex program, a feasible point is an optimal solution iff it lies in a hyperplane with a normal vector belonging to the subdifferential of the objective function at this point. In several cases, the solution set of a variational inequality problem is shown to coincide with the solution set of a convex program with its dual gap function as objective function, while the mapping involved can be used to express the above normal vectors.

The Adaptive Projected Subgradient Method over the Fixed Point Set of Strongly Attracting Nonexpansive Mappings

This paper presents an algorithmic solution, the adaptive projected subgradient method, to the problem of asymptotically minimizing a certain sequence of non-negative continuous convex functions over the fixed point set of a strongly attracting nonexpansive mapping in a real Hilbert space. The method generalizes Polyak's subgradient algorithm for the convexly constrained minimization of a fixed nonsmooth function. By generating a strongly convergent and asymptotically optimal point sequence, the proposed method not only offers unifying principles for many projection-based adaptive filtering algorithms but also enhances the adaptive filtering methods with the set theoretic estimation's armory by allowing a variety of a priori information on the estimandum in the form, for example, of multiple intersecting closed convex sets.

ON CONNECTIONS BETWEEN DELTA-CONVEX MAPPINGS AND CONVEX OPERATORS

AbstractWe study conditions under which every delta-convex (d.c.) mapping is the difference of two continuous convex operators, and vice versa. In particular, we prove that each d.c. mapping $F:(a,b)\to Y$ is the difference of two continuous convex operators whenever $Y$ belongs to a large class of Banach lattices which includes all $L^{p}(\mu)$ spaces ($1\leq p\leq\infty$). The proof is based on a result about Jordan decomposition of vector-valued functions. New observations on Jordan decomposition of finitely additive vector-valued measures are also presented.

A note on geometrical properties of Banach spaces using ψ-direct sums

Let X be a Banach space and ψ a continuous convex function on [ 0 , 1 ] satisfying certain conditions. Let X ⊕ ψ X be the ψ-direct sum of X. In this note, we characterize the strict convexity, uniform convexity and uniformly non-squareness of Banach spaces using ψ-direct sums, which extends the well-known characterization of these spaces.

On Gâteaux Differentiability of Convex Functions in WCG Spaces

AbstractIt is shown, using the Borwein–Preiss variational principle that for every continuous convex function f on a weakly compactly generated space X, every x0 ∈ X and every weakly compact convex symmetric set K such that , there is a point of Gâteaux differentiability of f in x0 +K. This extends a Klee's result for separable spaces.

ON THE PRODUCT OF GÂTEAUX DIFFERENTIABILITY LOCALLY CONVEX SPACES

A locally convex space is said to be a Gâteaux differentiability space (GDS) provided every continuous convex function defined on a nonempty convex open subset D of the space is densely Gâteaux differentiable in D. This paper shows that the product of a GDS and a family of separable Fréchet spaces is a GDS, and that the product of a GDS and an arbitrary locally convex space endowed with the weak topology is a GDS.

The strong conical hull intersection property for convex programming

The strong conical hull intersection property (CHIP) is a geometric property of a collection of finitely many closed convex intersecting sets. This basic property, which was introduced by Deutsch et al. in 1997, is one of the central ingredients in the study of constrained interpolation and best approximation. In this paper we establish that the strong CHIP of intersecting sets of constraints is the key characterizing property for optimality and strong duality of convex programming problems. We first show that a sharpened strong CHIP is necessary and sufficient for a complete Lagrange multiplier characterization of optimality for the convex programming model problem **where C is a closed convex subset of a Banach space X, S is a closed convex cone which does not necessarily have non-empty interior, Y is a Banach space, ** is a continuous convex function and g:X→Y is a continuous S-convex function. We also show that the strong CHIP completely characterizes the strong duality for partially finite convex programs, where Y is finite dimensional and g(x)=−Ax+b and S is a polyhedral convex cone. Global sufficient conditions which are strictly weaker than the Slater type conditions are given for the strong CHIP and for the sharpened strong CHIP.

Measure of Non-Radon–Nikodym Property and Differentiability of Convex Functions on Banach Spaces

We introduce and discuss measure of non-Radon–Nikodym property to investigate differentiability of continuous convex functions on Banach spaces. Using this new concept, we establish some results about Asplund spaces and generic Frechet differentiability of continuous convex functions on non-Asplund spaces.

On nonlinear wave equations with degenerate damping and source terms

In this article we focus on the global well-posedness of the differential equation u t t − Δ u + | u | k ∂ j ( u t ) = | u | p − 1 u in Ω × ( 0 , T ) u_{tt}- \Delta u + |u|^k\partial j(u_t) = |u|^{ p-1}u \, \text { in } \Omega \times (0,T) , where ∂ j \partial j is a sub-differential of a continuous convex function j j . Under some conditions on j j and the parameters in the equations, we obtain several results on the existence of global solutions, uniqueness, nonexistence and propagation of regularity. Under nominal assumptions on the parameters we establish the existence of global generalized solutions. With further restrictions on the parameters we prove the existence and uniqueness of a global weak solution. In addition, we obtain a result on the nonexistence of global weak solutions to the equation whenever the exponent p p is greater than the critical value k + m k+m , and the initial energy is negative. We also address the issue of propagation of regularity. Specifically, under some restriction on the parameters, we prove that solutions that correspond to any regular initial data such that u 0 ∈ H 2 ( Ω ) ∩ H 0 1 ( Ω ) u_0\in H^2(\Omega )\cap H^1_0(\Omega ) , u 1 ∈ H 0 1 ( Ω ) u_1 \in H^1_0(\Omega ) are indeed strong solutions.

Porosity and differentiability in smooth Banach spaces

We improve a result of Preiss, Phelps and Namioka, showing that every submonotone mapping in a Gateaux smooth Banach space is single-valued on the complement of a $\sigma$-cone porous subset. If a Banach space $E$ has a uniformly $\beta$-differentiable Lipschitz bump function (with respect to some bornology $\beta$), then we show with a much simpler argument (localization of $\delta$-minimum of a perturbed function) that every continuous convex function on $E$ is $\beta$-differentiable on the complement of a $\sigma$-uniformly porous set.